Mechanical Proof of the Maxwell-Boltzmann Speed Distribution With Numerical Iterations

The Maxwell-Boltzmann speed distribution is the probability that describes speeds of particles ideal gases. valid for both un-mixed (one type particle) and mixed (two types particles). For particles, follow distribution. Also, most probable inversely proportional to square root mass.
 
 This paper proves ratio using computer-generated data based on Newton’s law motion. To achieve this, this derives density function ψ^ab(u_a;v_a,v_b)  of u_a particle with mass M_a after collision two in v_a M_b v_b.  is obtained through a unique procedure considers (1) randomness relative direction before by an angle α. (2) another independent β.
 used equation below numerical iterations get new distributions P_new^a(u_a) from old P_old^a(v_a), repeat P_old^a(v_a)=P_new^a(v_a), where n_a fraction M_a. P_new^1(u_1)=n_1 ∫_0^∞ ψ^11(u_1;v_1,v’_1) P_old^1(v_1) P_old^1(v’_1) dv_1 dv’_1
                           +n_2 ψ^12(u_1;v_1,v_2) P_old^2(v_2) dv_2
 P_new^2(u_2)=n_1 ψ^21(u_2;v_2,v_1) dv_2 dv_1
 ψ^22(u_2;v_2,v’_2) P_old^2(v’_2) dv’_2
 final converge distributions. Moreover, root-mean-square masses as predicted Avogadro’s law.